1. Field of the Invention
The present invention relates to an integral value calculating device and a function gradient calculating device for easily calculating a transport coefficient, etc. by indicating the graph for the data outputted by a molecular dynamics simulator prior to an analysis of the data so that the scope of a calculation can be easily specified.
Molecular dynamics simulators have been developed as application technologies of super-computers, etc. capable of performing arithmetic operations for various advanced scientific technologies. These simulators figure out the properties, etc. of substances by simulating the behaviors of a molecule according to a molecular dynamics method after receiving the initial coordinates and an initial speed of an atom, an inter-atom potential function, the mass of an atom, the electric charge of an atom, etc.
Before describing the prior art of a molecular dynamics simulator pertaining to the analysis of output data, the summary of a molecular dynamics method is briefly explained as follows.
A molecular dynamics method is a computer simulation technology for analyzing the properties of multi-particle substances consisting of a number of particles by moving the particles according to the kinetic rules of the classical dynamics. The method itself originated in the old days, and a number of new technologies for the method have been developed since 1980, thus extending the scope of objects to be obtained by these technologies. This greatly depends on the development of the method of simulating a case under condition of fixed temperature and pressure while in the old days various experiments were carried out under condition of fixed energies and volume.
In computational physics used in the development of new substances (materials), the non-empirical molecular orbital method is in the stage of practical use in developing a fine molecular material having a small number of electrons. In this field, calculating, composing, and property-measuring processes have been used in combination as a new style of developing molecules. However, the classical molecular dynamics method in which potential functions are used for interatomic force and intermolecular force acts as an important part in designing materials for an atomic and molecular multi-particles body consisting of hundreds, thousands, or possibly tens of thousands of atoms where it is hard to successfully apply electronics even with the capabilities of giga flops realized by the latest computers.
Before the development of computer simulations, statistical mechanics had been used in analyzing according to micro data such as the correlation among atoms, etc. the structures and properties of a number of atomic and molecular groups. However, if the correlation among atoms is complicated to some extent, then exact solutions of basic equations in statistical mechanics cannot be obtained. Meanwhile, molecular dynamics realizes a simulation technology in which a multi-particles system Newton equation is analytically solved and the information similar to that obtained by statistical mechanics can be provided.
FIG. 1 shows input and output information according to the molecular dynamics method. Potential functions indicating the correlation among atoms and molecules, and physical environmental conditions such as temperatures and pressure, etc. are applied as the input information. Under these conditions, multi-particles system Newton equations are solved, and the resultant coordinates of the positions of atoms at predetermined points are statistically processed to provide thermodynamics properties (internal energies, specific heat, elastic constants, etc.) as the output information. If the coordinates of the position of an atom and its speed at each point are statistically processed, then dynamic properties (diffusion coefficients, shear viscosity, electric conductivity, thermal conductivity, etc.) and spectroscopic properties can be provided as the output information.
In a simulation based on the above described molecular dynamics, a substance such as a liquid, for example, composed of the enormous number of molecules is processed by tracing the movement of some of the molecules through a computer simulation so as to obtain the macroscopic properties of the substance. Then, conducting a simulation under condition of fixed temperature and pressure, for example, enables the structural phase change, where the particle positions in a crystal and the form of the crystal are subject to changes, to be directly detected. Necessarily, the number of particles and the time length of analysis greatly depend on the computational capabilities of computer systems, and an important object of this simulation technology has been to efficiently obtaining the macroscopic properties of substances using the smallest possible number of particle systems.
Self-diffusion coefficient, shear viscosity, bulk viscosity, thermal conductivity, electric conductivity, etc. can be obtained as transport coefficients by analyzing the data outputted by a simulator based on the molecular dynamics method. At this time, an error may be bigger, or the calculation time may be prolonged depending on the range of applicable output data. Therefore, it is required that the calculation range should be appropriately informed.
Conventionally, there are two calculation methods, that is, a method using a time correlation function and a method using a mean square displacement, when a transport coefficient is calculated using data obtained by the molecular dynamics method.
As well-known as a linear application theory, there is the following correlation between the time differentiation W'(t) of a certain dynamics volume W(t) and the transport coefficient K of the volume transported by W(t). ##EQU1##
W'(t) can be calculated by the molecular dynamics method, but time t is limited and the integral calculation is carried out for the limited range. Therefore, in the practical calculation, the transport coefficient K is approximated in the following equation. ##EQU2##
The precision greatly depends on the integral range.
The integral range generates the following graph. EQU F(t)=&lt;[W(t)W'(0)]&gt;
(where &lt; &gt; indicates and hereinafter refers to an average sample)
The value is effective up to the time t.sub.0 where the amplitude of the function F(t) has been converged sufficiently. This is the method of calculating a transport coefficient using a time correlation function.
The transport coefficient K can also be obtained as follows. ##EQU3##
because the following equation can exist mathematically. ##EQU4##
In this case, the dynamics volume W(t) is calculated by the molecular dynamics method to draw a graph G(t)=&lt;[W(t)-W(0)].sup.2 &gt;, and the value, of the gradient of the linear portion in the graph. This is the method of calculating a transport coefficient using a mean square displacement.
Conventionally, when the range of data used for obtaining such a transport coefficient is designated, a trial-and-error test is carried out based on empirical knowledge, or an appropriate range is determined by manually drawing an approximate graph.
Just for information, equations used in calculating a transport coefficient using data outputted by the calculation according to the molecular dynamics method are explained briefly as follows.
As described above, there is the following correlation among a certain dynamics volume W(t), its time differentiation W'(t), and the transport coefficient K of the volume transported by W(t). ##EQU5## The correlation among various transport coefficients K, dynamics volume W(t), and the time differentiation W'(t) is represented as follows.
(1) Self-diffusion coefficient D: EQU K=D EQU W(t)=vec-r.sub.i (t)=(x.sub.i (t), y.sub.i (t), z.sub.i (t)) EQU W'(t)=vec-v.sub.i (t)=(v.sub.ix (t), v.sub.iy (t), v.sub.iz (t))
where vec-r.sub.i (t) indicates a position vector of a particle i at the time t, and vec-v.sub.i (t) indicates a speed vector of a particle i at the time t.
v.sub.ix (t), v.sub.iy (t), and v.sub.iz (t) indicate time differentiations of x.sub.i (t), y.sub.i (t), z.sub.i (t) respectively.
(2) Shear viscosity .eta.: EQU K=Vk.sub.B T.eta.
(where V indicates volume, T temperature, and k.sub.B Boltzmann's constant) ##EQU6##
where .SIGMA. indicates a sum of i=1 through N when the number of particles is N.
m indicates the mass of a particle, F.sub.ix (t), F.sub.iy (t), and F.sub.iz (t) indicate respectively x, y, and z components of the force applied to the particle i.
(3) Bulk viscosity .xi.: ##EQU7##
(where .SIGMA. indicates the sum of i=1 through N (the number of particles))
(4) Thermal conductivity .lambda.: ##EQU8##
where .phi..sub.i (t) indicates the potential energy of a particle i at the time t.
(.SIGMA. indicates the sum of i=1 through N (the number of particles))
(5) Electric conductivity .sigma.: EQU K=Vk.sub.B T.sigma. EQU W(t)=.SIGMA.Q.sub.i vec-r.sub.i (t) EQU W'(t)=.SIGMA.Q.sub.i vec-v.sub.i (t)
where Q.sub.i indicates the electric charge of the particle i.
(.SIGMA. indicates the sum of i=1 through N (the number of particles))
According to the molecular dynamics method, the coordinates, speed, potential energy, etc. of each particle at each point are calculated, the results of which are used in calculating W(t) and W'(t) by the above mentioned equations.
As described above, the integral range should be determined when a transport coefficient is calculated using a time correlation function. However, it is not efficient to calculate a transport coefficient by drawing the graph F(t)=&lt;[W'(t)W'(t)]&gt; on paper, etc., reading the time t.sub.0 where the amplitude of the function F(t) has been sufficiently converged, and applying the integral result to an executable program t.sub.0.
When a transport coefficient is calculated using a mean square displacement, the graph for Function G(t)=&lt;[W(t)-W(0)].sup.2 &gt; should be draught, the range of the linear portion be extracted, and then the gradient of the linear portion be read. Furthermore, the performance becomes more inefficient if the method of least square is adopted in calculating the gradient of the linear portion.
If unknown data in time series obtained by a computer system are analyzed, not limited to the case where a transport coefficient is calculated, then the range of the calculation must be determined depending on the types of the data so as to obtain a necessary result. Conventionally, there are no guidelines for selecting the range of the calculation. Accordingly, it should be selected by the empirical knowledge from a list of numerical character strings, or by manually drawing a graph. This often causes errors and requires considerable labor and time.